In this section, we obtain the existence of a local strong solution for (REF ) in the most general case, when the nonlinearities \(f\)  and \(g\)  satisfy the conditions [C1]-[C3]. As follows from Lemma REF , \(W_A(t,x) \in D({\mathbb {A}})\)  provided (REF ) holds. Thus by Lemma 5.13 [1]} we have \(W_A(t) = \int _{0}^{t} {\mathbb {A}}W_A(s) ds + W(t)\) . We proceed with the change of variables \(U = u - W_A(t)\) . This way, the pair \((U,y)\)  solves
\({\left\lbrace \begin{array}{ll}dU = (-{\mathbb {A}}U - f(U+ W_A(t),y) + I(t))dt;\\dw=-g(U+W_A(t),w)dt.\end{array}\right.}\) 
